We need to take a step back from this question to see the bigger picture. First, we must define 'core' training. I'll keep it simple, and define it as simply improving the stability (timing and activation) of the muscles of the torso (including the deep muscles).
Next, we need to get an idea of how the 'core' is part of the running technique. What role does it play? How significant is that role?
Stabilizing the 'core' seems to be a component of the more complex running movement.
According to the chart, isolated joint mobility and stability are the foundation for basic movement patterns, and complex movement patterns, like running. That brings us to part of the answer: a minimum level of joint stability is needed for effectively and efficiently performing basic and complex movements.
So, if someone lacks an adequate level of 'core' stability, then 'core' training will have a positive carry-over to a complex movement pattern, like running.
If the opposite is true, and someone has adequate 'core' stability, then 'core' training will not have a positive carry-over to running. The answer to the question for this person is 'no.'
Hopefully, this explanation sheds some light on HTS training philosophy. Step one, make sure you have adequate (not maximal) joint mobility and stability. If any joint(s) aren't up to snuff, address this (especially beginners and those with injuries). Next, basic movement patterns (such as: pressing, lifting, carrying, pulling, squatting, lunging, etc.) require adequate joint mobility and stability. Basic movement patterns integrate mutiple joints across multiple planes. They provide more bang for your buck, and provide more carry-over to (specific) complex movement patterns.